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O Level · Mathematics 0580

Prime Factors, HCF & LCM

Revise Prime Factors, HCF & LCM for Mathematics 0580 (O Level) — revision notes and instant AI marking.

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1. Prime Factors & Factor Trees

Every number can be broken down into prime numbers that multiply together. This is called prime factor decomposition, and it's one of the most powerful tools in number theory.

Prime Factor: A prime number that divides exactly into a given number.

Example: The prime factors of 30 are 2, 3, and 5 because 2 × 3 × 5 = 30

📌 What is a Prime Number?

Prime Number: A number with exactly 2 factors: itself and 1.

First 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

📌 Method 1: Factor Tree

A factor tree breaks a number into pairs of factors until all branches end in prime numbers.

Steps:

1️⃣ Start with the number at the top

2️⃣ Split it into any two factors (not necessarily prime)

3️⃣ Split each non-prime factor into two more factors

4️⃣ Continue until all branches end in prime numbers (circle these)

5️⃣ Collect all the circled primes at the bottom

Example: Prime Factorization of 432

432 = 2 × 216
= 2 × 2 × 108
= 2 × 2 × 2 × 54
= 2 × 2 × 2 × 2 × 27
= 2 × 2 × 2 × 2 × 3 × 9
= 2 × 2 × 2 × 2 × 3 × 3 × 3
= 2⁴ × 3³

📌 Writing the Answer Using Indices/Powers

Instead of writing 2 × 2 × 2 × 2 × 3 × 3 × 3, use powers:

432 = 2⁴ × 3³

This is much cleaner! Notice: 2 appears 4 times (so 2⁴) and 3 appears 3 times (so 3³).

🧠 Key Rules:

Prime factorization is unique – any number breaks down the same way no matter which factors you choose first
Always write prime factors in ascending order (smallest to largest)
Use indices/powers to show repeated factors
The product of the prime factors always equals the original number

Q1: Express 60 as a product of its prime factors using indices.

Q2: If 144 = 2⁴ × 3², verify this is correct by multiplying it out.

2. Identifying Square & Cube Numbers

Prime factor decomposition reveals hidden patterns. If all the powers are even, it's a perfect square. If all are multiples of 3, it's a perfect cube.

📌 Perfect Square Numbers

Square Number: If ALL the powers in the prime factorization are EVEN, the number is a perfect square.

Example: 7056 = 2⁴ × 3² × 7²
All powers (4, 2, 2) are even, so 7056 is a perfect square.
Actually: 7056 = (2² × 3 × 7)² = 84²

📌 Perfect Cube Numbers

Cube Number: If ALL the powers in the prime factorization are MULTIPLES OF 3, the number is a perfect cube.

Example: 1728000 = 2⁹ × 3³ × 5³
All powers (9, 3, 3) are multiples of 3, so 1728000 is a perfect cube.
Actually: 1728000 = (2³ × 3 × 5)³ = 120³

Example: Is 360 a perfect square?

360 = 2³ × 3² × 5¹
The power of 2 is 3 (odd) ✗
The power of 5 is 1 (odd) ✗
No, 360 is NOT a perfect square.

Q1: Is 1000 a perfect cube? (Hint: 1000 = 2³ × 5³)

Q2: Is 72 = 2³ × 3² a perfect square?

3. Finding Square Roots Using Prime Factors

Prime factors make finding square roots easy – just halve all the powers!

📌 Square Root of a Perfect Square

Method:

1️⃣ Write the number as a product of prime factors with powers

2️⃣ Halve all the powers

3️⃣ Multiply out the result

Example: Find √144

144 = 2⁴ × 3²
Halve the powers: √(2⁴ × 3²) = 2² × 3¹
Calculate: 4 × 3 = 12
√144 = 12

📌 Exact Square Root of Non-Perfect Squares

If it's not a perfect square, we can still find the exact square root by separating odd and even powers.

Method (for non-perfect squares):

1️⃣ Write as prime factors: 1440 = 2⁵ × 3² × 5

2️⃣ Rewrite odd powers as even + 1: 2⁵ = 2⁴ × 2¹, 5¹ = 5¹

3️⃣ Group: 1440 = 2⁴ × 3² × (2 × 5)

4️⃣ Take square root: √1440 = 2² × 3 × √10 = 12√10

Example: Find the exact square root of 1440

1440 = 2⁵ × 3² × 5
Rewrite: 1440 = 2⁴ × 2 × 3² × 5
√1440 = 2² × 3 × √(2 × 5)
= 4 × 3 × √10
= 12√10 (exact form)

Even and odd powers separated!
The even powers (2⁴, 3²) come out.
The odd powers (2, 5) stay under the root.

🧠 Key Pattern:

Even power out → divide power by 2
Odd power stays in radical → it becomes √(prime × prime)
√(2 × 5) = √10, √(2 × 2) = 2, √(3²) = 3

Q1: Find √(2⁴ × 3² × 5²)

Q2: Find the exact square root of 200 = 2³ × 5²

4. Highest Common Factor (HCF)

HCF is the largest number that divides evenly into two or more numbers. It's used to simplify fractions.

HCF (Highest Common Factor): The largest number that divides exactly into all given numbers.

Example: HCF(24, 36) = 12 because 12 divides both numbers and is the largest such number.

📌 Method 1: Using Prime Factors

Multiply the LOWEST power of each COMMON prime factor.

Steps:

1️⃣ Write both numbers as products of prime factors

2️⃣ Find which primes appear in BOTH numbers

3️⃣ For each common prime, pick the LOWEST power

4️⃣ Multiply these together to get the HCF

Example: Find HCF(24, 60)

24 = 2³ × 3¹
60 = 2² × 3¹ × 5¹

Common primes: 2 and 3
Lowest power of 2: 2² (from 60)
Lowest power of 3: 3¹ (same in both)
5 is NOT common

HCF = 2² × 3 = 4 × 3 = 12

📌 Method 2: Using Venn Diagram

Put common factors in the centre, unique factors on the sides. HCF = product of centre only.

🧠 HCF Rule:

HCF uses ONLY common factors
Use the LOWEST power of each common prime
Multiply them together
HCF could be 1 (if no common primes)
HCF could be one of the numbers itself!

Q1: Find HCF(36, 120) where 36 = 2² × 3² and 120 = 2³ × 3 × 5

Q2: Find HCF(42, 90) where 42 = 2 × 3 × 7 and 90 = 2 × 3² × 5

5. Lowest Common Multiple (LCM)

LCM is the smallest number that appears in the times tables of two or more numbers. It's used to add fractions.

LCM (Lowest Common Multiple): The smallest number that is a multiple of all given numbers.

Example: LCM(6, 9) = 18 because 18 is in both the 6 times table (6×3) and 9 times table (9×2).

📌 Method: Using Prime Factors (Most Efficient)

Multiply the HIGHEST power of EVERY prime that appears in either number.

Steps:

1️⃣ Write both numbers as products of prime factors

2️⃣ List EVERY prime that appears in either number

3️⃣ For each prime, pick the HIGHEST power (from either number)

4️⃣ Multiply these together to get the LCM

Example: Find LCM(72, 540)

72 = 2³ × 3²
540 = 2² × 3³ × 5¹

Every prime: 2, 3, 5
Highest power of 2: 2³ (from 72)
Highest power of 3: 3³ (from 540)
Highest power of 5: 5¹ (only in 540)

LCM = 2³ × 3³ × 5 = 8 × 27 × 5 = 1080

🧠 LCM Rule:

LCM uses EVERY prime from BOTH numbers
Use the HIGHEST power of each prime
Even if a prime is only in one number, include it!
Multiply them together
LCM could be one of the numbers itself!

📌 Key Difference: HCF vs LCM

Feature	HCF	LCM
What primes?	COMMON only	ALL primes
Which power?	LOWEST	HIGHEST
Size?	Usually smaller	Usually larger
Use for...	Simplifying fractions	Finding common denominators

Q1: Find LCM(36, 120) where 36 = 2² × 3² and 120 = 2³ × 3 × 5

Q2: Find LCM(12, 18) where 12 = 2² × 3 and 18 = 2 × 3²

6. Worked Example – Complete Problem

Problem: Given N = 2³ × 3² × 5⁷ and AN = B where A is an integer and B is a non-zero square number. Find the smallest value of A.

📌 Solution

Step 1: Understand what we need

For B to be a square number, ALL powers in AN must be EVEN.

Step 2: Substitute and multiply

                            AN = A(2³ × 3² × 5⁷)

                            This must equal B (a perfect square)

Step 3: Find what A needs

Currently: 2³ (odd) ✗, 3² (even) ✓, 5⁷ (odd) ✗
A must have: 2¹ (to make 2³ × 2¹ = 2⁴), and 5¹ (to make 5⁷ × 5¹ = 5⁸)

Step 4: Calculate the smallest A

                            A = 2 × 5 = 10

                            Check: AN = 10(2³ × 3² × 5⁷) = (2×5)(2³ × 3² × 5⁷) = 2⁴ × 3² × 5⁸ ✓

                            All powers even!

Answer: A = 10

📋 Chapter Summary

Prime factor decomposition is the foundation for HCF, LCM, and understanding the structure of numbers. Master this, and you'll tackle fraction problems, simplification, and algebraic factoring with ease.

🔢 Prime Factorization

Every number breaks down uniquely into prime factors. Use factor trees, write with powers (indices), always ascending order.

⬆️ Square & Cube Numbers

All even powers = perfect square. All powers divisible by 3 = perfect cube. Use this to identify and find roots.

↓ HCF (Highest Common Factor)

Common primes only, lowest powers, multiply together. Used to simplify fractions and find GCD.

↑ LCM (Lowest Common Multiple)

All primes from both numbers, highest powers, multiply together. Used for common denominators.

🧠 What to Memorise

The Prime Numbers (First 15)

✏️ Primes to Know:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
2 is the ONLY even prime
All others are odd

Key Formulas & Processes

✏️ Prime Factorization

Always write in ASCENDING order
Use INDICES/POWERS for repeated factors
Unique for every number

✏️ Square Numbers (All Powers EVEN)

√(even powers) = divide each power by 2
Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100...

✏️ Cube Numbers (All Powers ÷ 3)

Perfect cubes: 1, 8, 27, 64, 125, 216, 343...
All powers must be multiples of 3

✏️ HCF: COMMON PRIMES, LOWEST POWERS

HCF(24, 36) where 24=2³×3, 36=2²×3²
Common: 2 and 3
Lowest powers: 2² × 3¹ = 12
Simplifying fractions: divide both by HCF

✏️ LCM: ALL PRIMES, HIGHEST POWERS

LCM(24, 36) where 24=2³×3, 36=2²×3²
All primes: 2 and 3
Highest powers: 2³ × 3² = 72
Common denominators: use LCM

✏️ The Relationship: HCF × LCM = Product

HCF(a,b) × LCM(a,b) = a × b
Check your work with this formula!

✅ Concepts Checklist

Tick off each concept as you master it. Return to sections where you're unsure.

I can identify prime numbers
I can draw a factor tree correctly
I can express a number as a product of prime factors
I can use indices/powers in my answer
I can identify perfect square numbers using powers
I can identify perfect cube numbers using powers
I can find the square root of a perfect square using powers
I can find the exact square root of non-perfect squares (e.g., 12√10)
I understand the definition of HCF
I can find HCF using prime factors (common, lowest powers)
I understand the definition of LCM
I can find LCM using prime factors (all primes, highest powers)
I know the difference between HCF and LCM (common vs all, lowest vs highest)
I can use HCF to simplify fractions
I can use LCM to find common denominators
I understand the relationship HCF × LCM = product
I can solve problems involving finding missing factors (like AN = B)
I can explain why the answer to a factor problem is unique

🎯 Exam Tips & Common Mistakes

📌 Common Mistakes to Avoid

❌ Mistake 1: Forgetting to use indices/powers ✓ Correct: 432 = 2⁴ × 3³ (simplified form)
Not: 432 = 2 × 2 × 2 × 2 × 3 × 3 × 3 (clunky, loses marks)

❌ Mistake 2: Not writing primes in ascending order ✓ Correct: 60 = 2² × 3 × 5 (ascending)
Not: 60 = 5 × 3 × 2² (wrong order, looks careless)

❌ Mistake 3: Confusing HCF and LCM ✓ Correct: HCF = common primes, LOWEST powers (usually smaller)
LCM = ALL primes, HIGHEST powers (usually larger)

❌ Mistake 4: Using wrong powers in HCF/LCM ✓ Correct for HCF(24, 60) where 24=2³×3, 60=2²×3×5: Pick LOWEST: 2² (not 2³) and 3¹ (same in both)
HCF = 2² × 3 = 12 ✓

❌ Mistake 5: Forgetting non-common primes in LCM ✓ Correct for LCM(24, 60): Include 5 even though it's only in 60!
LCM = 2³ × 3 × 5 = 120 ✓
Not just 2³ × 3 = 24 ✗

❌ Mistake 6: Not showing working for factor trees ✓ Correct: Show each step of splitting. Examiners award marks for method, not just the answer.

❌ Mistake 7: Halving wrong powers for square roots ✓ Correct for √(2⁴ × 3²): Halve EACH power: 2⁴ becomes 2², and 3² becomes 3¹
Result: 2² × 3 = 12 ✓

📌 Exam Strategy Tips

💡 Tip 1: Always Show Your Factor Tree Even if you can see the answer, the working is worth marks. Show each split clearly.

💡 Tip 2: Check with HCF × LCM = Product After finding HCF and LCM, multiply them. Should equal the product of the original numbers.

💡 Tip 3: Circle All Prime Numbers in Your Factor Tree Makes it easy to spot and count them. Less chance of missing one.

💡 Tip 4: Use Venn Diagrams for HCF/LCM Questions Draw circles, put common factors in centre. HCF = centre only. LCM = everything.

💡 Tip 5: Always Simplify Using Indices in Final Answer No credit for 2 × 2 × 2 × 3 × 3 when 2³ × 3² is available. Use indices.

💡 Tip 6: For Square Root Problems, Separate Even & Odd Powers Even powers come out of the radical. Odd powers stay inside (with one factor out).

📌 Question Types You'll See

Type 1: "Express X as a product of its prime factors"

Draw factor tree → show working → write using indices
Example: 432 = 2⁴ × 3³

Type 2: "Is X a perfect square/cube?"

Find prime factorization → check powers
All even = square ✓ | All ÷3 = cube ✓

Type 3: "Find HCF(A, B)"

Prime factorize both → common primes only → lowest powers → multiply

Type 4: "Find LCM(A, B)"

Prime factorize both → all primes → highest powers → multiply

Type 5: "Find √(perfect square)"

Prime factorize → halve all powers → multiply them out

Type 6: "Find exact √(non-perfect square)"

Prime factorize → separate even/odd powers → simplify

📌 Final Pre-Exam Checklist

Can I quickly draw a factor tree and circle the primes?
Do I remember to write answers using indices?
Can I spot the difference between HCF and LCM in a question?
Do I know the rule: HCF = common, lowest; LCM = all, highest?
Can I use the check: HCF × LCM = product?
Have I practised at least 3 factor trees from start to finish?
Can I separate even and odd powers for exact square roots?

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